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Positive solution of a problem of Emden-Fowler type with a free boundary. (English) Zbl 0637.34013

Let us consider the question of existence of a solution of the problem: Find \(T>0\) and \(y\in C[0,T]\cap C^ 1[0,T)\cap C^ 2(0,T)\) such that \[ (1)\quad y''(t)+q(t)y(t)^{\gamma}=0,\quad t\in (0,T),\quad y(0)=y(T)=0,\quad y'(0)=\alpha,\quad y(t)>0,\quad t\in (0,T), \] where \(\gamma\geq 1\), \(\alpha >0\) and q, a positive function, are given. If \(q(t)=t^{\beta}\), \(\beta\) a real number, (1) reduces to the Emden- Fowler equation whose origin lies in theories concerning gaseous dynamics in astrophysics. Under the additional assumptions:
i) \(q\in C^ 2(0,+\infty)\), \(\forall t>0:\) \(q(t)>0\), \(t\to t^{\gamma}\), q(t) belongs to \(L^ 1(0,1).\)
ii)\(\eta\),\(\eta\) ”\(\in L^ 1(1,+\infty)\), where \(\eta \in C^ 2(0,+\infty)\) is defined by \(\eta (t)=[q(t)]^{-1/\gamma +3}\) and \(\int^{+\infty}_{1}[\eta (t)]^{-2} dt=+\infty.\)
iii) If \(\gamma >1:\) \[ \lim_{t\downarrow 0}t^{-1} \eta (t)[\int^{+\infty}_{t}\eta (s)| \eta ''(s)| ds]^{2/\gamma - 1}=0, \] the author proves that the problem (1.1) has a unique solution (T,y). If \(q(t)=t^{\beta}\), it is shown that (1) has a solution if and only if \(\gamma +2\beta +3>0\). Moreover a monotone iteration scheme holds for the approximation of a positive solution.
Reviewer: A.Cañada

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
34A45 Theoretical approximation of solutions to ordinary differential equations
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